The modular curve $X_{15}$

Curve name $X_{15}$
Index $6$
Level $8$
Genus $0$
Does the subgroup contain $-I$? Yes
Generating matrices $ \left[ \begin{matrix} 1 & 1 \\ 0 & 1 \end{matrix}\right], \left[ \begin{matrix} 7 & 0 \\ 2 & 1 \end{matrix}\right], \left[ \begin{matrix} 5 & 0 \\ 2 & 1 \end{matrix}\right]$
Images in lower levels
LevelIndex of imageCorresponding curve
$2$ $3$ $X_{6}$
$4$ $3$ $X_{6}$
Meaning/Special name Elliptic curves with a cyclic $4$-isogeny defined over $\mathbb{Q}(\sqrt{-2})$
Chosen covering $X_{6}$
Curves that $X_{15}$ minimally covers $X_{6}$
Curves that minimally cover $X_{15}$ $X_{41}$, $X_{46}$
Curves that minimally cover $X_{15}$ and have infinitely many rational points. $X_{41}$, $X_{46}$
Model \[\mathbb{P}^{1}, \mathbb{Q}(X_{15}) = \mathbb{Q}(f_{15}), f_{6} = 8f_{15}^{2} + 48\]
Info about rational points None
Comments on finding rational points None
Elliptic curve whose $2$-adic image is the subgroup $y^2 + xy + y = x^3 - 6x + 4$, with conductor $66$
Generic density of odd order reductions $5123/21504$

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